Finite difference scheme for longitudinal dispersion in open channels

The available analytical and numerical solutions of the equation for longitudinal dispersion in open channels are limited to uniform flows. Presented in this paper is a solution technique based on combined operator approach where advection and diffusion processes of longitudinal dispersion equation are treated concurrently in non-uniform flows. A variable size spatial grid based finite difference solution of the advection process has been obtained by developing a variable spatial grid so that the root of the trajectory of the concentration characteristic passes through the computational nodes. For solution of the diffusion process, Crank-Nicholson scheme has been used. To eliminate the possibility of numerical oscillations, weighting coefficient has been introduced to the pollutant concentration in the time stepping. Proof-of-the-concept tests have been made using the existing numerical and analytical solutions as the basis. The model has been extended by incorporating in it the one-dimensional grid search method for determination of D L values using observed temporal variation of concentration (C-t curves) at two or more stations. Finally, a procedure of computing the C-t curves at downstream locations is presented in the paper.